Copyright
Copyright © 2018 by Scott E. Page
Cover design by Chin-Yee Lai
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CONTENTS

Cover
Title Page

Copyright
Dedication
Epigraph
Prologue
1 The Many-Model Thinker
2 Why Model?
3 The Science of Many Models
4 Modeling Human Actors
5 Normal Distributions: The Bell Curve
6 Power-Law Distributions: Long Tails
7 Linear Models
8 Concavity and Convexity
9 Models of Value and Power
10 Network Models
11 Broadcast, Diffusion, and Contagion
12 Entropy: Modeling Uncertainty
13 Random Walks
14 Path Dependence
15 Local Interaction Models
16 Lyapunov Functions and Equilibria
17 Markov Models
18 Systems Dynamics Models
19 Threshold Models with Feedbacks
20 Spatial and Hedonic Choice


21 Game Theory Models Times Three
22 Models of Cooperation
23 Collective Action Problems
24 Mechanism Design

25 Signaling Models
26 Learning Models
27 Multi-Armed Bandit Problems
28 Rugged-Landscape Models
29 Opioids, Inequality, and Humility
About the Author
Notes
Bibliography
Index


To Michael D. Cohen
(1945–2013)


It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic
elements as simple and as few as possible without having to surrender the adequate
representation of a single datum of experience.
—ALBERT EINSTEIN


Prologue
To me success means effectiveness in the world, that I am able to carry my ideas and values into
the world—that I am able to change it in positive ways.
—Maxine Hong Kingston
This book began as the result of a chance meeting with Michael Cohen in 2005 near the flower garden
in the mall adjacent to the University of Michigan’s West Hall. Michael, a scholar known for his
generosity, made a comment that altered my teaching career. With a twinkle in his eyes, Michael said,
“Scottie, I once taught a course called Introduction to Modeling for Social Scientists, based on a book
written by Charles Lave and James March. You should resurrect the course. It needs you.”

It needed me? I returned to my office a little confused, so I chased down an old course syllabus. I
discovered that Michael had misled me. The course did not need me. I needed it. I had been wanting
to develop a course that would introduce students to the core ideas of complex systems—networks,
diversity, learning, large events, path dependence, tipping points—that would be relevant to their
daily lives and future careers. By teaching modeling, I could make students better thinkers while
introducing them to complexity. I could teach them tools that would improve their abilities to reason,
explain, predict, design, communicate, act, and explore
The course’s motivating idea would be that we must confront the complexity of the modern world
with multiple models. At semester’s end, rather than see the world from a particular angle, students
would see the world through many lenses. They would be standing in houses with many windows,
able to look in multiple directions. My students would be better prepared for the complex challenges
before them—improving education, reducing poverty, creating sustainable growth, finding meaningful
work in an age of artificial intelligence, managing resources, and designing robust financial,
economic, and political systems.
The next fall, I resurrected the course. I contemplated rebranding it as Thirty-Two Models That
Will Turn You into a Genius, but the culture at Michigan frowns on hyperbole, so I stuck with
Michael’s title: An Introduction to Modeling. Lave and March’s book proved to be a brilliant
introduction. However, modeling had made huge advances in the intervening decades. I needed an
updated version that included models of long-tailed distributions, networks, rugged landscapes, and
random walks. I needed a book that discussed complexity.
So I began to write. For two years, the ground proved rocky. My plow moved at a slow place.
One spring day, I again ran into Michael, this time in the arch-way underneath West Hall. I had been
questioning the course, which was now drawing twenty students. Were models too abstract for
undergraduates? Should I teach a different course on a specific issue or policy domain? Michael
offered up a smile, noting that any endeavor worth pursuing merited questioning. As we parted,
Michael commented on the importance and value of helping people think clearly. He told me not to
give up, that he took joy in my challenges.
In the fall of 2012, the ground under the course shifted. Vice Provost Martha Pollack asked me to
teach an online version—what is now called a MOOC. With a tablet computer, a $29 camera, and a



$90 microphone, Model Thinking was born. With assistance from too many people at Michigan,
Coursera, and Stanford University to thank properly (a quick shout-out to Tom Hickey, who did
yeoman’s work), I reorganized my lectures into a form suitable for an online course, dividing each
subject into modules and removing all copyrighted material. With my dog Bounder as an audience, I
taped and retaped lectures.
The first offering of Model Thinking drew 60,000 students. That number now approaches a
million. The popularity of the online course led me to abandon the book. I thought the project
unnecessary, but, over the next two years, my email inbox began to fill with requests for a book to
complement the online lectures. Then Michael Cohen lost his battle with cancer, and I felt that I
needed to finish the book. I reopened the manuscript folder.
Writing a book requires large blocks of time and spaces that allow for clear thought. The poet
Wallace Stevens wrote, “Perhaps the truth depends on a walk around the lake.” I relied on a close
analog: mind-clearing swims across Winans Lake, where my family spends our summers. Throughout
the writing process, the continuous life I share with the love of my life, Jenna Bednar, our sons, Orrie
and Cooper, and our enormous dogs, Bounder, Oda, and Hildy, has brought laughter, comfort, and
opportunities—among them Orrie having one week to correct the penultimate draft’s mathematical
errors and Jenna having two weeks to identify instances of unclear writing, logical flaws, and
muddled thinking. As has been true of most of my written work, this manuscript might be best
described as an original draft by Scott Page with substantial revision by Jenna Bednar.
During the seven-year period of writing this book, my children have transitioned from pre-teens to
young adults. Orrie is now off to college. Cooper follows next year. In the interval between sketching
the initial outline and submitting the final version, my family has consumed copious amounts of
bibimbap, pasta carbonara, and oatmeal chocolate chip cookies, taken the saws and loppers to scores
of fallen branches and limbs, repaired dozens of breaks in the backyard fence, embarked on numerous
failed initiatives to reduce the entropy in the basement and garage, and wished and hoped for the ice
on the lake to be suitable for skating. We have also had to accept loss. Midway through the project,
my mother, Marilyn Tamboer Page, died from a sudden heart attack while enjoying the bliss of her
routine daily walk with her dog. Not a day goes by when I do not reflect on the love she showered on
her family and the support she gave to others.

The book before you is as complete as it can be at this moment in time. Doubtless, new models
will be created, and old models will find new uses creating gaps in this current offering. As I humbly
send the manuscript out into the world, I feel that my efforts will have been repaid if you, the reader,
find the models and ideas within to be useful and generative, and that you are able to carry them out
into the world and change it in positive ways.
If one day, when sitting in some professor’s or graduate student’s office, preferably at a college or
university in my beloved Midwest, I scan the bookshelves and find this book leaning, as it has during
its writing, on a well-worn copy of Lave and March, then my efforts will have been all the sweeter.


1. The Many-Model Thinker
To become wise you’ve got to have models in your head. And you’ve got to array your experience
—both vicarious and direct—on this latticework of models.
—Charlie Munger
This is a book about models. It describes dozens of models in straightforward language and explains
how to apply them. Models are formal structures represented in mathematics and diagrams that help
us to understand the world. Mastery of models improves your ability to reason, explain, design,
communicate, act, predict, and explore.
This book promotes a many-model thinking approach: the application of ensembles of models to
make sense of complex phenomena. The core idea is that many-model thinking produces wisdom
through a diverse ensemble of logical frames. The various models accentuate different causal forces.
Their insights and implications overlap and interweave. By engaging many models as frames, we
develop nuanced, deep understandings. The book includes formal arguments to make the case for
multiple models along with myriad real-world examples.
The book has a pragmatic focus. Many-model thinking has tremendous practical value. Practice it,
and you will better understand complex phenomena. You will reason better. You exhibit fewer gaps
in your reasoning and make more robust decisions in your career, community activities, and personal
life. You may even become wise.
Twenty-five years ago, a book of models would have been intended for professors and graduate
students studying business, policy, and the social sciences along with financial analysts, actuaries,

and members of the intelligence community. These were the people who applied models and, not
coincidentally, they were also the people most engaged with large data sets. Today, a book of models
has a much larger audience: the vast universe of knowledge workers, who, owing to the rise of big
data, now find working with models a part of their daily lives.
Organizing and interpreting data with models has become a core competency for business
strategists, urban planners, economists, medical professionals, engineers, actuaries, and
environmental scientists among others. Anyone who analyzes data, formulates business strategies,
allocates resources, designs products and protocols, or makes hiring decisions encounters models. It
follows that mastering the material in this book—particularly the models covering innovation,
forecasting, data binning, learning, and market entry timing—will be of practical value to many.
Thinking with models will do more than improve your performance at work. It will make you a
better citizen and a more thoughtful contributor to civic life. It will make you more adept at evaluating
economic and political events. You will be able to identify flaws in your logic and in that of others.
You will learn to identify when you are allowing ideology to supplant reason and have richer, more
layered insights into the implications of policy initiatives, whether they be proposed greenbelts or
mandatory drug tests.
These benefits will accrue from an engagement with a variety of models—not hundreds, but a few


dozen. The models in this book offer a good starting collection. They come from multiple disciplines
and include the Prisoners’ Dilemma, the Race to the Bottom, and the SIR model of disease
transmission. All of these models share a common form: they assume a set of entities—often people
or organizations—and describe how they interact.
The models we cover fall into three classes: simplifications of the world, mathematical analogies,
and exploratory, artificial constructs. In whatever form, a model must be tractable. It must be simple
enough that within it we can apply logic. For example, we cover a model of communicable diseases
that consists of infected, susceptible, and recovered people that assumes a rate of contagion. Using the
model we can derive a contagion threshold, a tipping point, above which the disease spreads. We can
also determine the proportion of people we must vaccinate to stop the disease from spreading.
As powerful as single models can be, a collection of models accomplishes even more. With many

models, we avoid the narrowness inherent in each individual model. A many-models approach
illuminates each component model’s blind spots. Policy choices made based on single models may
ignore important features of the world such as income disparity, identity diversity, and
interdependencies with other systems.1 With many models, we build logical understandings of
multiple processes. We see how causal processes overlap and interact. We create the possibility of
making sense of the complexity that characterizes our economic, political, and social worlds. And,
we do so without abandoning rigor—model thinking ensures logical coherence. That logic can be then
be grounded in evidence by taking models to data to test, refine, and improve them. In sum, when our
thinking is informed by diverse logically consistent, empirically validated frames, we are more likely
to make wise choices.

Models in the Age of Data
The appearance of a book on models may seem out of place in the era of big data. Today, data exists
in unprecedented dimensionality and granularity. Customer purchase data, which used to arrive in
monthly aggregates on printed paper, now streams instantaneously with geospatial, temporal, and
consumer tags. Student academic performance data now includes scores on every homework, paper,
quiz, and exam, as opposed to semester-end summary grades. In the past, a farmer might mention dry
ground at a monthly Grange meeting. Now, tractors transmit instantaneous data on soil conditions and
moisture levels in square-foot increments. Investment firms track dozens of ratios and trends for
thousands of stocks and use natural-language processing tools to parse documents. Doctors can pull
up page upon page of individual patient records that can include relevant genetic markers.
A mere twenty-five years ago, most of us had access to little more than a few bookshelves’ worth
of knowledge. Perhaps your place of work had a small reference library, or at home you had a
collection of encyclopedias and a few dozen reference books. Academics and government and
private-sector researchers had access to large library collections, but even they had to physically
visit the material. As late as the turn of the millennium, academics could be found shuttling back and
forth between card catalog rooms, microfiche collections, library stacks, and special collections in
search of information.
That has all changed. Content that had been paper-bound for centuries now flows in tiny packets
through the air. So too does the information about the here and now. News that arrived on our



doorsteps on newsprint once a day now flows in a continuous digital stream into our personal
devices. Stock prices, sports scores, and news of political events and cultural happenings can all be
accessed with a swipe or query.
As impressive as the data may be, it is no panacea. We now know what has happened and is
happening, but, owing to the complexity of the modern world, we may be less capable of
understanding why it happened. Empirical findings may be misleading. Data on piece-rate work often
shows that the more people are paid per unit of output, the less they produce. A model in which pay
depends on work conditions can explain those data. If conditions are poor so that producing output is
difficult, per unit pay may be high. If conditions are good, per unit pay may be low. Thus, higher pay
does not lead to less productivity. Instead, more difficult work conditions require higher per unit
pay.2
In addition, most of our social data—that is, data about our economic, social, and political
phenomena—documents only moments or intervals in time. It rarely tells us universal truths. Our
economic, social, and political worlds are not stationary. Boys may outscore girls on standardized
tests in one decade and girls may outscore boys the next. The reasons people vote today may differ
from the reasons they vote in coming decades.
We need models to make sense of the fire-hose-like streams of data that cross our computer
screens. Thus, it is because we have so much data that this might also be called the age of many
models. Look across the academy, government, the business world, and the nonprofit sector, and you
struggle to find a domain of inquiry or decision not informed by models. Consulting giants McKinsey
and Deloitte build models to formulate business strategies. Financial firms such as BlackRock and
JPMorgan Chase apply models to select investments. Actuaries at State Farm and Allstate use models
to calibrate risk when pricing insurance policies. The people team at Google builds predictive
analytic models to evaluate its more than three million job applicants. College and university
admissions officers construct predictive models to select from among tens of thousands of applicants.
The Office of Management and Budget constructs economic models to predict the effects of tax
policies. Warner Brothers applies data analytics to create models of audience responses. Amazon
develops machine learning models to make product recommendations. Researchers funded by the

National Institutes of Health build mathematical models of human genomics to search for and evaluate
potential cures for cancer. The Gates Foundation uses epidemiological models to design vaccination
strategies. Even sports teams use models to evaluate draft prospects and trade opportunities and to
formulate within-game strategies. By relying on models to select players and strategies, the Chicago
Cubs won a World Series championship after more than a century of failures.
To people who use models, the rise of model thinking has an even simpler explanation: models
make us smarter. Without models, people suffer from a laundry list of cognitive shortcomings: we
overweight recent events, we assign probabilities based on reasonableness, and we ignore base rates.
Without models, we have limited capacity to include data. With models, we clarify assumptions and
think logically. And, we can leverage big data to fit, calibrate, and test causal and correlative claims.
With models, we think better. In head-to-head competitions between models and people, models
win.3

Why We Need Many Models


In this book we advocate using not just one model in a given situation but many models. The logic
behind the many-model approach builds on the age-old idea that we achieve wisdom through a
multiplicity of lenses. This idea traces back to Aristotle, who wrote of the value of combining the
excellences of many. A diversity of perspectives was also a motivation for the great-books
movement, which collected 102 important transferable ideas in The Great Ideas: A Syntopicon of
Great Books of the Western World . The approach finds a modern voice in the work of Maxine Hong
Kingston, who wrote in The Woman Warrior , “I learned to make my mind large, as the universe is
large, so that there is room for paradoxes.” It is also the basis for pragmatic actions in the world of
business and policy. Recent books argue that if we want to understand of international relations, we
should not model the world exclusively as a group of self-interested nations with well-defined
objectives, or only as an evolving nexus of multinational corporations and intergovernmental
organizations. We should do both.4
As commonsensical as the many-model approach may seem, keep in mind that it runs counter to
how we teach models and the practice of modeling. The traditional approach—the one taught in high

school—relies on a one-to-one logic: one problem requires one model. For example: now we apply
Newton’s first law; now we apply the second; now the third. Or: here we use the replicator equation
to show the size of the rabbit population in the next period. In this traditional approach, the objective
is to (a) identify the one proper model and (b) apply it correctly. Many-model thinking challenges that
approach. It advocates trying many models. Had you used many-model thinking in ninth grade, you
might have been held back. Use it now, and you will move forward.
Academic papers, for the most part, follow the one-to-one approach as well, even though they use
those single models to explain complex phenomena: Trump voters in the 2016 election were those
who had been left behind economically. Or: the quality of a child’s second-grade teacher determines
how economically successful that child will be as an adult.5 A stream of best-selling nonfiction titles
present cures for our ills based on single-model thinking: Educational success depends on grit.
Inequality results from concentrations of capital. Our nation’s poor health is due to sugar
consumption. Each of these models may be true, but none is comprehensive. To confront the
complexity of these challenges, to create a world of broader educational achievement, will require
lattices of models.
By learning the models in this book, you can begin to build your own lattice. The models originate
from a broad spectrum of disciplines, addressing phenomena as varied as the causes of income
inequality, the distribution of power, the spread of diseases and fads, the conditions that precede
social uprisings, the evolution of cooperation, the emergence of order in cities, and the structure of
the internet. The models vary in their assumptions and their structure. Some describe small numbers
of rational, self-interested actors. Others describe large populations of rule-following altruists. Some
describe equilibrium processes. Others produce path dependence and complexity. The models also
differ in their uses. Some help predict and explain. Others guide actions, inform designs, or facilitate
communication. Still others create artificial worlds for our minds to explore.
The models share three common characteristics: First, they simplify, stripping away unnecessary
details, abstracting from reality, or creating anew from whole cloth. Second, they formalize, making
precise definitions. Models use mathematics, not words. A model might represent beliefs as
probability distributions over states of the world or preferences as rankings of alternatives. By
simplifying and making precise, they create tractable spaces within which we can work through logic,



generate hypotheses, design solutions, and fit data. Models create structures within which we can
think logically. As Wittgenstein wrote in his Tractatus Logico-Philosophicus, “Logic takes care of
itself; all we have to do is to look and see how it does it.” The logic will help to explain, predict,
communicate, and design. But the logic comes at a cost, which leads to their third characteristic: all
models are wrong, as George Box noted.6 That is true of all models; even the sublime creations of
Newton that we refer to as laws hold only at certain scales. Models are wrong because they simplify.
They omit details. By considering many models, we can overcome the narrowing of rigor by
crisscrossing the landscape of the possible.
To rely on a single model is hubris. It invites disaster. To believe that a single equation can
explain or predict complex real-world phenomena is to fall prey to the charisma of clean, spare
mathematical forms. We should not expect any one model to produce exact numerical predictions of
sea levels in 10,000 years or of unemployment rates in 10 months. We need many models to make
sense of complex systems. Complex systems like politics, the economy, international relations, or the
brain exhibit ever-changing emergent structures and patterns that lie between ordered and random. By
definition, complex phenomena are difficult to explain, evolve, or predict.7
Thus, we confront a disconnect. On the one hand, we need models to think coherently. On the other
hand, any single model with a few moving parts cannot make sense of high-dimensional, complex
phenomena such as patterns in international trade policy, trends in the consumer products industry, or
adaptive responses within the brain. No Newton can write a three-variable equation that explains
monthly employment, election outcomes, or reductions in crime. If we hope to understand the spread
of diseases, variation in educational performance, the variety of flora and fauna, the effect of artificial
intelligence on job markets, the impact of humans on the earth’s climate, or the likelihood of social
uprisings, we must come at them with machine learning models, systems dynamics models, game
theory models, and agent-based models.

The Wisdom Hierarchy
To sketch the argument for many-model thinking, we begin with a query from poet and dramatist T. S.
Eliot: “Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in
information?” To that we might add, where is the information we have lost in all this data?

Eliot’s questioning can be formalized as the wisdom hierarchy. At the bottom of the hierarchy lie
data: raw, uncoded events, experiences, and phenomena. Births, deaths, market transactions, votes,
music downloads, rainfall, soccer matches, and speciation events. Data can be long strings of zeros
and ones, time stamps, and linkages between pages. Data lack meaning, organization, or structure.
Information names and partitions data into categories. Examples clarify the distinction between
data and information. Rain falling on your head is data. Total rainfall for the month of July in
Burlington, Vermont, and Lake Ontario’s water level are information. The bright red peppers and
yellow corn on farmers’ stands surrounding the capitol in Madison, Wisconsin, on market Saturdays
are data. The farmers’ total sales are information.


Figure 1.1: How Models Transform Data into Wisdom

We live in an age of abundant information. A century and a half ago, knowing information brought
great economic and social status. Jane Austen’s Emma asks if Frank Churchill is “a young man of
information.” Today she would not care. Churchill, like everyone else, would have a smartphone. The
question is whether he could put that information to use. As Fyodor Dostoyevsky writes in Crime and
Punishment, “We’ve got facts, they say. But facts aren’t everything; at least half the battle consists in
how one makes use of them!”
Plato defined knowledge as justified true belief. More modern definitions refer to it as
understandings of correlative, causal, and logical relationships. Knowledge organizes information.
Knowledge often takes model form. Economic models of market competition, sociological models of
networks, geological models of earthquakes, ecological models of niche formation, and psychological
models of learning all embed knowledge. Those models explain and predict. Models of chemical
bonds explain why metallic bonds prevent us from putting our hands through steel doors while
hydrogen bonds yield to our weight when we dive into a lake.8
Atop the hierarchy lies wisdom, the ability to identify and apply relevant knowledge. Wisdom
requires many-model thinking. Sometimes, wisdom consists of selecting the best model, as if drawing
from a quiver of arrows. Other times, wisdom can be achieved by averaging models; this is common
when making predictions. (We discuss the value of model averaging in the next section.) When taking

actions, wise people apply multiple models like a doctor’s set of diagnostic tests. They use models to
rule out some actions and privilege others. Wise people and teams construct a dialogue across
models, exploring their overlaps and differences.
Wisdom can consist of selecting the correct knowledge or model; consider the following physics
problem: A small stuffed cheetah falls from an airplane’s hold at 20,000 feet. How much damage will
it do upon landing? A student might know a gravity model and a terminal velocity model. The two
models give different insights. The gravity model predicts that the stuffed animal would tear through a
car’s roof. The terminal velocity model predicts that the toy cheetah’s speed tops out at around 10
mph.9 Wisdom consists of knowing to apply the terminal velocity model. A person could stand on the
ground and catch the soft cheetah in her hands. To quote the evolutionary biologist J. B. S. Haldane,
“You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a
slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a
horse splashes.”
In the stuffed-cheetah problem, arriving at the correct solution requires information (the weight of
the toy), knowledge (the terminal velocity model), and wisdom (selecting the correct model).
Business and policy leaders also rely on information and knowledge to make wise choices. On
October 9, 2008, the value of Iceland’s currency, the króna, began a free fall. Eric Ball, then treasurer
of software giant Oracle, was faced with a decision. A few weeks prior he had dealt with the


domestic repercussions of the home mortgage crisis. Iceland’s situations posed an international
concern. Oracle held billions of dollars in overseas assets. Ball considered network contagion
models of financial collapse. He also thought of economic models of supply and demand in which the
magnitude of a price change correlates with the size of the market shock. In 2008, Iceland had a GDP
of $12 billion, or less than six months’ revenues for McDonald’s Corporation. Ball recollected
thinking, “Iceland is smaller than Fresno. Go back to work.”10 The key to understanding this event,
and many-model thinking generally, lies in recognizing that Ball did not search among many models to
find one that supported an action that he had already decided to take. He did not use many models to
find one that justified his action. Instead, he evaluated two models as possibly useful and then chose
the better one. Ball had the right information (Iceland is small), chose the right model (supply and

demand), and made a wise choice.
We next show how to create a dialogue among multiple models by reconsidering two historical
events: the 2008 global financial market collapse, which reduced total wealth (or what had been
thought to be wealth) by trillions of dollars, resulting in a four-year global recession, and the 1961
Cuban missile crisis, which nearly resulted in nuclear war.
The 2008 financial collapse has multiple explanations: too much foreign investment, overleveraged investment banks, lack of oversight in the mortgage approval process, blissful optimism
among home-flipping consumers, the complexity of financial instruments, a misunderstanding of risk,
and greedy bankers who knew the bubble existed and expected a bailout. Superficial evidence aligns
with each of these accounts: money flowed in from China, loan originators wrote toxic mortgages,
investment banks had high leverage ratios, financial instruments were too complex for most to
understand, and some banks expected a bailout. With models we can adjudicate between these
accounts and check the internal consistency of these accounts: Do they make logical sense? We can
also calibrate the models and test the magnitude of the effects.
The economist Andrew Lo, exercising many-model thinking, evaluates twenty-one accounts of the
crisis. He finds each to be lacking. It does not make sense that investors would contribute to a bubble
that they knew would lead to a global crisis. Hence, the extent of the bubble must have been a
surprise to many. Financial firms may well have assumed the other firms had done due diligence
when in fact they had not. Second, what were, in retrospect, clearly toxic (low-quality) bundles of
mortgages found buyers. Had global collapse been a foregone conclusion, the buyers would not have
existed. And while leverage ratios had increased since 2002, they were not much higher than they had
been in 1998. And as for the notion that the government would bail out the banks, Lehman Brothers
collapsed on September 15, 2008; with over $600 billion in holdings, it was the largest bankruptcy in
US history. The government did not intervene.
Lo finds that each account contains a logical gap. The data, such as it is, privileges no single
explanation. As Lo summarizes: “We should strive at the outset to entertain as many interpretations of
the same set of objective facts as we can, and hope that a more nuanced and internally consistent
understanding of the crisis emerges in the fullness of time.” He goes on to say, “Only by collecting a
diverse and often mutually contradictory set of narratives can we eventually develop a more complete
understanding of the crisis.” No single model suffices.11
In Essence of Decision, Graham Allison undertakes a many-model approach to explain the Cuban

missile crisis. On April 17, 1961, a CIA-trained paramilitary group landed on the shores of Cuba in a
failed attempt to overthrow Fidel Castro’s communist regime, increasing tensions between the United


States and the Soviet Union, Cuba’s ally. In response, Soviet premier Nikita Khrushchev moved
short-range nuclear missiles to Cuba. President John F. Kennedy responded by blockading Cuba. The
Soviet Union backed down, and the crisis ended.
Allison interprets events with three models. He applies a rational-actor model to show that
Kennedy had three possible actions: start a nuclear war, invade Cuba, or impose a blockade. He
chose the blockade. The rational-actor model assumes that Kennedy draws a game tree with each
action followed by the possible responses by the Soviets. Kennedy then thinks through the Soviets’
optimal response. If, for example, Kennedy launched a nuclear attack, the Soviets would strike back,
resulting in millions dead. If Kennedy imposed a blockade, he would starve the Cubans. The Soviet
Union could either back down or launch missiles. Given that choice, the Soviet Union should back
down. The model reveals the central strategic logic at play and provides a rationale for Kennedy’s
bold choice to blockade Cuba.
Like all models, though, it is wrong. It ignores relevant details, allowing it to initially appear a
better explanation than it really is. The model neglects to add a stage in which the Soviets put the
missiles in Cuba. If the Soviets had been rational, they should have drawn the same tree as Kennedy
and realized that they would have to remove the missiles. The rational-actor model also fails to
explain why the Soviets did not hide the missiles.
Allison applies an organizational process model to explain these inconsistencies. A lack of
organizational capacity explains the Soviets’ failure to hide the missiles. The same model can explain
Kennedy’s choice to blockade. At the time, the United States Air Force lacked the capacity to wipe
out the missiles in a single strike. If even a single missile remained, it could kill millions of
Americans. Allison deftly combines the two models. An insight from the organizational model
changes the payoffs in the rational-choice model.
Allison adds a governmental process model. The other two models reduce countries to their
leaders: Kennedy acts for the United States and Khrushchev for the Soviet Union. The government
process model recognizes that Kennedy had to contend with Congress and that Khrushchev needed to

maintain a political base of support. Thus, Khrushchev’s placing of the missiles in Cuba signaled
strength.
Allison’s book shows the power of models alone and in dialogue. Each model clarifies our
thinking. The rational-actor model identifies possible actions once the missiles have arrived and
allows us to see the implications of those actions. The organizational model draws our attention to the
fact that organizations, not individuals, carry out those actions. The governmental process model
highlights the political cost of invasion. By evaluating events through all three lenses, we gain a
broader and deeper understanding. All models are wrong; many are useful.
In both examples, the different models explicate distinct causal forces. Multiple models can also
focus on different scales. In an oft-repeated tale, a child claims that the Earth rests on the back of a
giant elephant. A scientist asks the child what the elephant stands on, to which the child replies, “A
giant turtle.” Anticipating what’s about to come next, the child quickly adds, “Don’t even ask. It’s
turtles all the way down.”12 If the world were turtles all the way—if the world were self-similar—
then a model of the top level would apply at every level. But the economy, the political world, and
society are not turtles all the way down, nor is the brain. At the sub-micron level, the brain is made
up of molecules that form synapses, which in turn form neurons. The neurons combine in networks.
The networks overlap in elaborate ways that can be studied with brain imaging. These neuronal


networks exist on a scale below that of functional systems such as the cerebellum. Given that the
brain differs at each level, we need multiple models, and those models differ. The models that
characterize the robustness of neuronal networks bear little resemblance to the molecular biology
models used to explain brain cell function, which in turn differ from the psychological models used to
explain cognitive biases.
The success of many-model thinking depends on a degree of separability. In analyzing the 2008
financial crisis, we rely on separate models of foreign purchases of assets, of the bundling of assets,
and of increased leverage ratios. Allison drew implications from the game theoretic model without
considering the organizational model. In studying the human body, doctors separate the skeletal,
muscular, limbic, and nervous systems. That said, many-model thinking does not require that these
distinct models divide the system into independent parts. Confronted with a complex system, we

cannot, to paraphrase Plato, carve the world at its joints. We can partially isolate the major causal
threads and then explore how they are interwoven. In doing so, we will find that the data produced by
our economic, political, and social systems exhibits coherence. Social data is more than sequences of
incomprehensible hairballs that might have been spit up by the family cat.

Summary and Outline of the Book
To summarize, we live in a time awash in information and data. The same technological advances
generating those data shrink time and distance. They make economic, political, and social actors more
agile, capable of responding to economic and political events in an instant. They also increase
connectedness, and therefore complexity. We face a technologically induced paradox: we know more
about the world, but that world is more complex. In light of that complexity, any single model will be
more likely to fail. We should not though abandon models. To the contrary, we should privilege
logical coherence over intuition and double, triple, and even quadruple down on models and become
many-model thinkers.
Becoming a many-model thinker requires learning multiple models of which we gain a working
knowledge; we need to understand the formal descriptions of the models and know how to apply
them. We need not be experts. Hence, this book balances accessibility and depth. It can function both
as a resource and as a guide. The formal descriptions are isolated in stand-alone boxes. It avoids line
after line of equations, which overwhelm even the most dedicated readers. The formalism that
remains should be engaged and absorbed. Modeling is a craft, mastered through engagement; it is not
a spectator sport. It requires deliberate practice. In modeling, mathematics and logic play the role of
an expert coach. They correct our flaws.
The remainder of the book is organized as follows: Chapters 2 and 3 motivate the many-model
approach. Chapter 4 discusses the challenges of modeling people. The next twenty or so chapters
cover individual models or classes of models. By considering one type of model at a time, we can
better wrap our heads around its assumptions, implications, and applications. This structure also
means that we can pull the book from our bookshelves or open it in our browsers and find selfcontained analyses of linear models, prediction models, network models, contagion models, and
models of long-tailed distributions, learning, spatial competition, consumer preferences, path
dependence, innovation, and economic growth. Interspersed throughout the chapters are applications



of many-model thinking to a variety of problems and issues. The book concludes with two deeper
dives into the opioid epidemic and income inequality.


2. Why Model?
Knowing reality means constructing systems of transformations that correspond, more or less
adequately, to reality.
—Jean Piaget
In this chapter, we define types of models. Models are often described as simplifications of the
world. They can be, but models can also take the form of analogies or be fictional worlds mined for
ideas and insights. We also describe the uses of models. In school, we apply models to explain data.
In practice, we can also use models to predict, design, and take actions. We can use models to
explore ideas and possibilities. And we can use models to communicate ideas and understandings.
The value of models also resides in their ability to reveal conditions under which results hold.
Most of what we know holds only in some cases: the square of the longest side of a triangle equals
the sum of the squares of the other sides only if the longest side is opposite a right angle. Models
reveal similar conditions for our intuitions. With models we can parse out when diseases spread,
when markets work, when voting leads to good outcomes, and when crowds make accurate
predictions. None of those is a sure thing.
This chapter consists of two parts. In the first, we describe the three types of models. In the
second, we cover the uses of models: to reason, explain, design, communicate, act, predict, and
explore. These form the acronym REDCAPE, a notso-subtle reminder that many-model thinking
endows us with superpowers.1

Types of Models
When constructing a model, we take one of three approaches. We can aim for realism and follow an
embodiment approach. Such models include the important parts and either strip away unnecessary
dimensions and attributes or lump them together. Models of ecological glades, legislatures, and traffic
systems take this approach, as do climate models and models of the brain. Or we can take an analogy

approach and abstract from reality. We can model crime spreading like a disease and the taking of
political positions as choices on a left-right continuum. The spherical cow is a favorite classroom
example of the analogy approach: to make an estimate of the amount of leather in a cowhide, we
assume a spherical cow. We do so because the integral tables in the back of calculus textbooks
include tan(x) and cos(x) but not cow(x).2
While the embodiment approach stresses realism, the analogy approach tries to capture the
essence of a process, system, or phenomenon. When a physicist assumes away friction but otherwise
makes realistic assumptions, she takes the embodiment approach. When an economist represents
competing firms as different species and defines product niches, she makes an analogy. She does so
using a model developed to embody a different system. No bright line differentiates the embodiment


approach from the analogy approach. Psychological models of learning that assign weights to
alternatives lump together dopamine responses and other factors; they also invoke the analogy of a
scale on which we balance alternatives.
A third approach, the alternative reality approach, purposely does not represent or capture
reality. These models function as analytic and computational playgrounds in which we can explore
possibilities. This approach allows us to discover general insights that apply outside our physical and
social world. They help us to understand the implications of real-world constraints: What if energy
could be sent safely and efficiently through the air? And they allow us to run impossible experiments:
What if we tried to evolve a brain? This book contains a few such models, notably the Game of Life,
which consists of a checkerboard whose squares are classified as either alive (black) or dead (white)
that switch between alive and dead according to fixed rules. Though unrealistic, the model produces
insights into self-organization, complexity, and, some argue, even life itself.
Whether embodying a more complex reality, creating an analogy, or building a made-up world for
exploring ideas, a model must be communicable and tractable. We should be able to write the model
in a formal language such as mathematics or computer code. When describing a model, we cannot
toss out terms like beliefs or preferences without providing a formal description. Beliefs can be
represented as a probability distribution over a set of events or priors. Preferences can be
represented in several ways such as a ranking over a set of alternatives or as a mathematical function.

How tractable something is means how amenable it is to analysis. In the past, analysis relied on
mathematical or logical reasoning. A modeler had to be able to prove each step in an argument. This
constraint led to an aesthetic that valued stark models. English friar and theologian William of
Ockham (1287–1347) wrote, “Plurality must never be posited without necessity.” Einstein summed
up this principle, known as Ockham’s Razor, as follows: everything should be made as simple as
possible, but not simpler. Today, when we run up against the constraint of analytic tractability, we
can turn to computation. We can build elaborate models with many moving parts without concern for
analytic tractability. Scientists take this approach when constructing models of the global climate, the
brain, forest fires, and traffic. They still pay heed to Ockham’s advice, but recognize that “as simple
as possible” might require a lot of moving parts.

The Seven Uses of Models
The academic literature describes dozens of uses of models. Here, we focus on seven categories of
uses: to reason, explain, design, communicate, act, predict, and explore.

The Uses of Models (REDCAPE)
Reason: To identify conditions and deduce logical implications.
Explain: To provide (testable) explanations for empirical phenomena.
Design: To choose features of institutions, policies, and rules.


Communicate: To relate knowledge and understandings.
Act: To guide policy choices and strategic actions.
Predict: To make numerical and categorical predictions of future and unknown phenomena.
Explore: To investigate possibilities and hypotheticals.

REDCAPE: Reason
When constructing a model, we identify the most important actors and entities along with relevant
characteristics. We then describe how those parts interact and aggregate, enabling us to derive what
follows from what, and why. In doing so, we improve our reasoning. While what we can derive

depends upon what we assume, we uncover more than tautologies. Rarely can we infer the full range
of implications of our assumptions from inspection alone. We need formal logic. Logic also reveals
impossibilities and possibilities. With it, we can derive precise and sometimes unexpected
relationships. We can discover the conditionality of our intuitions.
Arrow’s theorem provides an example of how logic reveals impossibilities. The model addresses
the question of whether individual preferences aggregate to form a collective preference. This model
represents preferences as ordinal rankings over alternatives. If applied to five Italian restaurants,
denoted by the letters A through E, the model allows any of the 120 orderings. Arrow required that
the collective ordering be monotonic (if everyone ranks A above B, then so does the collective),
independent of irrelevant alternatives (if no person’s relative rankings of A and B are unchanged but
rankings of other alternatives change, then the order of A and B in the collective ranking does not
change), and nondictatorial (no single person should decide the collective ordering). Arrow then
proved that if any preferences are allowed, then no collective ordering necessarily exists.3
Logic can also reveal paradoxes. Using models we can show the possibility of each subpopulation
containing a larger percentage of women than men but the total population containing a larger
percentage of men, a phenomenon (Simpson’s paradox). This actually happened: 1973, the University
of California, Berkeley, accepted a larger percentage of women in most departments. Overall, it
accepted men at a higher rate. Models also show that it is possible for two losing bets, when played
alternately, to produce a positive expected return (Parrondo’s paradox). With models, we can show
that it is possible to add a node to a network and reduce the total length of the edges needed to
connect all the nodes.4
We should not dismiss these examples as mathematical novelties. Each has practical applications:
efforts to increase the population of women could backfire, combinations of losing investments could
win, and the total length of a network of electric lines, pipelines, ethernet lines, or roads could be be
reduced by adding more nodes.
Logic also uncovers mathematical relationships. Given Euclid’s axioms, a triangle can be
uniquely determined by any two angles and a side, or by any two sides and an angle. With standard
assumptions about consumer and firm behaviors, in markets with a large number of competing firms,
price equals marginal cost. Some results are unexpected: among them the friendship paradox, which



states that in any friendship network, on average, people’s friends have more friends than they do.
The paradox arises because highly popular people have more friends. Figure 2.1 shows Zachary’s
Karate Network. The person represented by the dark circle has six friends, denoted by gray circles.
His friends have nine friends on average. These people are represented by white circles. Over the
entire network, twenty-nine of the thirty-four people have friends who are more popular than they
are.5 Later we show that if we make a few more assumptions, most people’s friends will also be, on
average, better-looking, kinder, richer, and smarter than they are.

Figure 2.1: The Friendship Paradox: A Person’s Friends Have More Friends

Last, and most important of all, logic reveals the conditionality of truths. A politician may claim
that lowering income taxes increases government revenue by spurring economic growth. A
rudimentary model in which revenue equals the tax rate times the income level proves that revenue
increases only if the percentage growth in income exceeds the percentage cut in taxes.6 Thus, a 10%
cut in income taxes increases revenue only if it causes income to grow by more than 10%. The
politician’s logic only holds given certain conditions. Models identify those conditions.
The power of conditionality becomes evident when we contrast claims derived from models with
narrative claims, even when the latter have empirical support. Consider the management proverb first
things first: the idea that when facing multiple tasks, you should do the most important task first. This
rule is also known as big rocks first, because when filling a bucket with rocks of various sizes, you
should put the big rocks in first—if you put the little rocks in first, the big rocks will not fit.
The rule big rocks first, inferred from expert observation, may be a good rule most of the time, but
it is unconditional. A model-based approach would make specific assumptions about the task and then
derive an optimal rule. In the bin packing problem, a set of objects of various sizes (or weights) must
be allocated into bins of finite capacity. The objective is to use as few bins as possible. Imagine, for
example, you are packing up your apartment and putting everything into 2-foot-by-2-foot boxes.
Ordering your possessions by size and putting each object in the first box with sufficient space
(known as the first fit algorithm) turns out to be quite effective. Big rocks first works well.
However, suppose that we consider a more complex task: allocating space on the International Space

Station for research projects. Each project has a payload weight, a size, and power requirements
along with demands on the astronauts’ time and cognitive abilities. Each also makes a potential
scientific contribution. Even if we came up with some measure of bigness as a weighted average of
these attributes, big rocks first would prove a poor rule given the dimensionality of
interdependencies. More sophisticated algorithms and possibly market mechanisms would perform
much better.7 Thus, under some conditions, big rocks first is a good rule. Under other conditions, it is
not. With models, we can trace the boundaries of when we should place the big rocks first and when
we should not.


Critics of formalism claim that models repackage what we already know, that they pour old wine
into shiny mathematical bottles, that we do not need a model to know that two heads are better than
one or that he who hesitates is lost. We can learn the value of commitment from reading of Odysseus
tying himself to the mast. That criticism fails to recognize that inferences drawn from models take
conditional forms: if condition A holds, then result B follows (e.g., if you are packing bins and size is
the only constraint, pack the biggest objects first). Lessons drawn from literature or proverbial advice
from great thinkers often provide no conditions. If we try to lead our lives or manage others by
unconditional rules, we find ourselves lost in a sea of opposite proverbs. Are two heads better than
one? Or, do too many cooks spoil the broth?
Proverb: Two heads are better than one
Opposite: Too many cooks spoil the broth
Proverb: He who hesitates is lost
Opposite: A stitch in time saves nine
Proverb: Tie yourself to the mast
Opposite: Keep your options open
Proverb: The perfect is the enemy of the good
Opposite: Do it well or not at all
Proverb: Actions speak louder than words
Opposite: The pen is mightier than the sword
While opposite proverbs abound, opposite theorems cannot. Within models, we make assumptions

and prove theorems. Two theorems that disagree on the optimal action, make different predictions, or
offer distinct explanations must make different assumptions.

REDCAPE: Explain
Models provide clear logical explanations for empirical phenomena. Economic models explain price
movements and market shares. Physics models explain the rate of falling objects and the shape of
trajectories. Biological models explain the distributions of species. Epidemiological models explain
the speed and patterns of disease spread. Geophysical models explain the size distribution of
earthquakes.
Models can explain point values and changes in their values. A model can explain the current
price of pork belly futures and why prices rose over the past six months. A model can explain why a
president appoints a moderate Supreme Court justice and why a candidate moves to the left or right.
Models also explain shape: models of the diffusion of ideas, technologies, and diseases produce an
S-shaped curve of adoption (or contagion).
The models we learn in physics, such as Boyle’s Law (a model stating that the pressure of oxygen
times the volume equals a constant (PV = k)), explain phenomena unreasonably well.8 If we know the
volume, we can estimate the constant k, and then explain or predict pressure P as a function of V and


k. The model owes its accuracy to the fact that gases consist of simple parts that exist in large
numbers and follow fixed rules: any two oxygen molecules placed in the identical situation follow the
same physical laws. They exist in such large numbers that statistical averaging cancels out any
randomness. Most social phenomena share none of these three attributes: social actors are
heterogeneous, interact in small groups, and do not follow fixed rules. People also think. Even more
problematic, people respond to social influences, meaning that behavioral variations may not cancel
out. As a result, social phenomena are much less predictable than physical phenomena.9
The most effective models explain both straightforward outcomes and puzzling ones. Textbook
models of markets can explain why an unanticipated increase in the demand for a normal good like
shoes or potato chips increases the price in the short run, an intuitive result. These same models
explain why in the long run, demand increases have less of an effect on price than the marginal cost of

producing the good. Increases in demand can even produce reductions in price that result from
increased returns to scale in production, a more surprising result. The same models can explain
paradoxes such as why diamonds, which have little practical value, have high prices, but water, a
necessity for survival, costs little.
As for the claim that models can explain anything: it is true, they can. However, a model-based
explanation includes formal assumptions and explicit causal chains. Those assumptions and causal
chains can be taken to data. A model that claims that high levels of criminal behavior can be
explained by low probabilities of being caught can be tested.

REDCAPE: Design
Models aid in design by providing frameworks within which we can contemplate the implications of
choices. Engineers use models to design supply chains. Computer scientists use models to design web
protocols. Social scientists used models to design institutions.
In July 1993, a group of economists met at Caltech in Pasadena, California, to design an auction to
allocate the electronic spectrum for cellular phones. In the past, the government had allocated
spectrum rights to large companies for modest fees. A provision within the Omnibus Budget
Reconciliation Act of 1993 allowed for auctioning the spectrum to raise money.
The radio signal from a tower covers a geographic range. Therefore, the government sought to sell
licenses for specific regions: Western Oklahoma, Northern California, Massachusetts, Eastern Texas,
and so on. This created a design challenge. The value of any given license for a company depended
on the other licenses that company won. The license for Southern California would be worth more to
a company that also owned the license for Northern California, for example. Economists refer to
these interdependent valuations as externalities. The externalities had two main sources: construction
and advertising. Holding neighboring licenses meant lower construction costs and the potential to
exploit overlapping media markets.
The externalities created a problem with holding simultaneous auctions. A company trying to win
a bundle of licenses might lose one license to another bidder and therefore lose the externalities. That
company might then want to back out of its bids on other licenses. Sequential auctions had a different
shortcoming. Bidders would underbid in early auctions to hedge against losing subsequent licenses.
A successful auction design had to be immune to strategic manipulation, generate efficient

outcomes, and be comprehensible to participants. The economists used game theory models to